Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Topic: Integration by Substitution**
Not Attempted
**Instructions:**
Use the substitution method to determine the integral:
\[ \int (24x^3 + 5) \cdot \sqrt{6x^4 + 5x + 10} \, dx. \]
**Explanation:**
In this problem, you need to use the substitution method to solve the given integral. The substitution method is a technique used in calculus to simplify integrals by making a substitution for a part of the integrand, which makes it easier to integrate.
**Steps to Solve:**
1. **Identify a Substitution:** Look for an expression within the integral that, when differentiated, appears elsewhere in the integrand. In this case, consider \( u = 6x^4 + 5x + 10 \).
2. **Differentiate the Substitution:** Compute \( \frac{du}{dx} \) and rearrange to express \( dx \) in terms of \( du \).
3. **Change Variables:** Substitute the expressions for \( u \) and \( dx \) back into the integral.
4. **Integrate:** Perform the integration with respect to \( u \).
5. **Back-Substitute:** Replace \( u \) with the original variable to express the solution in terms of \( x \).
By following these steps, you will find the solution to the integral using the substitution method.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fce0e1f50-5655-4918-bba8-458441e12ec5%2F054ce00f-6ba1-4ba0-a859-f7d463c56282%2Fdiczqu_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Topic: Integration by Substitution**
Not Attempted
**Instructions:**
Use the substitution method to determine the integral:
\[ \int (24x^3 + 5) \cdot \sqrt{6x^4 + 5x + 10} \, dx. \]
**Explanation:**
In this problem, you need to use the substitution method to solve the given integral. The substitution method is a technique used in calculus to simplify integrals by making a substitution for a part of the integrand, which makes it easier to integrate.
**Steps to Solve:**
1. **Identify a Substitution:** Look for an expression within the integral that, when differentiated, appears elsewhere in the integrand. In this case, consider \( u = 6x^4 + 5x + 10 \).
2. **Differentiate the Substitution:** Compute \( \frac{du}{dx} \) and rearrange to express \( dx \) in terms of \( du \).
3. **Change Variables:** Substitute the expressions for \( u \) and \( dx \) back into the integral.
4. **Integrate:** Perform the integration with respect to \( u \).
5. **Back-Substitute:** Replace \( u \) with the original variable to express the solution in terms of \( x \).
By following these steps, you will find the solution to the integral using the substitution method.
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